Open course ware

Okay. This is lecture six in linear algebra, and we're at the start of this new chapter, chapter three in the text, which is really getting to the center of linear algebra. And I had time to make a first start on it at the end of lecture five. But now is lecture six is officially the lecture on vector spaces and subspaces. And then especially -- there are two subspaces that we're specially interested in. One is the column space of a matrix, the other is the null space of the matrix. So, I got to tell you what those are. Okay. So, first to remember from lecture five, what is a vector space? It's a bunch of vectors that -- where I'm allowed -- where I can add -- I can add any two vectors in the space and the answer stays in the space. Or I can multiply any vector in the space by any constant and the result stays in the space. So that's -- in fact if I combine those two into one, you can see that -- if I can add and I can multiply by numbers, that really ...

Okay. This is lecture five in linear algebra. And, it will complete this chapter of the book. So the last section of this chapter is two point seven that talks about permutations, which finished the previous lecture, and transposes, which also came in the previous lecture. There's a little more to do with those guys, permutations and transposes. But then the heart of the lecture will be the beginning of what you could say is the beginning of linear algebra, the beginning of real linear algebra which is seeing a bigger picture with vector spaces -- not just vectors, but spaces of vectors and sub-spaces of those spaces. So we're a little ahead of the syllabus, which is good, because we're coming to the place where, there's a lot to do. Okay. So, to begin with permutations. Can I just -- so these permutations, those are matrices P and they execute row exchanges. And we may need them. We may have a perfectly good matrix, a perfect matrix A that's invertibl...

KAREN: Okay. Today's our last class, and so, in this presentation, we hope to discuss and pull together a lot of different themes that we've talked about over the course of the semester, namely centered around discourses of power. So we've kind of teased apart the idea that there are political, economic, social, and linguistic discourses of power that impact the way people live their lives and how they identify, and hopefully by learning about these different discourses of power, we can find the treasure trove of understanding identity. RACHEL: So in terms of discourses of power, in the political sense, in our readings about the different issues in the Caribbean, we discussed the general topic was colonialization as a discourse of power and how colonialization, in all of these countries, established dominant discourses for racial, social, linguistic, and economic hierarchies of power. We-- and how these hierarchies kind of dictated the structure of the society...

Okay. This is the lecture on the singular value decomposition. But everybody calls it the SVD. So this is the final and best factorization of a matrix. Let me tell you what's coming. The factors will be, orthogonal matrix, diagonal matrix, orthogonal matrix. So it's things that we've seen before, these special good matrices, orthogonal diagonal. The new point is that we need two orthogonal matrices. A can be any matrix whatsoever. Any matrix whatsoever has this singular value decomposition, so a diagonal one in the middle, but I need two different -- probably different orthogonal matrices to be able to do this. Okay. And this factorization has jumped into importance and is properly, I think, maybe the bringing together of everything in this course. One thing we'll bring together is the very good family of matrices that we just studied, symmetric, positive, definite. Do you remember the stories with those guys? Because they were symmetric, their eigenvector...

Okay. This lecture is mostly about the idea of similar matrixes. I'm going to tell you what that word similar means and in what way two matrixes are called similar. But before I do that, I have a little more to say about positive definite matrixes. You can tell this is a subject I think is really important and I told you what positive definite meant -- it means that this -- this expression, this quadratic form, x transpose I x is always positive. But the direct way to test it was with eigenvalues or pivots or determinants. So I -- we know what it means, we know how to test it, but I didn't really say where positive definite matrixes come from. And so one thing I want to say is that they come from least squares in -- and all sorts of physical problems start with a rectangular matrix -- well, you remember in least squares the crucial combination was A transpose A. So I want to show that that's a positive definite matrix. Can -- so I -- I'm going to speak a l...